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Mirrors > Home > MPE Home > Th. List > Mathboxes > sepnsepolem2 | Structured version Visualization version GIF version |
Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 47030. Proof could be shortened by 1 step using ssdisjdr 46967. (Contributed by Zhi Wang, 1-Sep-2024.) |
Ref | Expression |
---|---|
sepnsepolem2.1 | β’ (π β π½ β Top) |
Ref | Expression |
---|---|
sepnsepolem2 | β’ (π β (βπ¦ β ((neiβπ½)βπ·)(π₯ β© π¦) = β β βπ¦ β π½ (π· β π¦ β§ (π₯ β© π¦) = β ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sepnsepolem2.1 | . 2 β’ (π β π½ β Top) | |
2 | id 22 | . . 3 β’ (π½ β Top β π½ β Top) | |
3 | sslin 4199 | . . . . 5 β’ (π§ β π¦ β (π₯ β© π§) β (π₯ β© π¦)) | |
4 | sseq0 4364 | . . . . . 6 β’ (((π₯ β© π§) β (π₯ β© π¦) β§ (π₯ β© π¦) = β ) β (π₯ β© π§) = β ) | |
5 | 4 | ex 414 | . . . . 5 β’ ((π₯ β© π§) β (π₯ β© π¦) β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
6 | 3, 5 | syl 17 | . . . 4 β’ (π§ β π¦ β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
7 | 6 | adantl 483 | . . 3 β’ ((π½ β Top β§ π§ β π¦) β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
8 | ineq2 4171 | . . . . 5 β’ (π¦ = π§ β (π₯ β© π¦) = (π₯ β© π§)) | |
9 | 8 | eqeq1d 2739 | . . . 4 β’ (π¦ = π§ β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
10 | 9 | adantl 483 | . . 3 β’ ((π½ β Top β§ π¦ = π§) β ((π₯ β© π¦) = β β (π₯ β© π§) = β )) |
11 | 2, 7, 10 | opnneieqv 47017 | . 2 β’ (π½ β Top β (βπ¦ β ((neiβπ½)βπ·)(π₯ β© π¦) = β β βπ¦ β π½ (π· β π¦ β§ (π₯ β© π¦) = β ))) |
12 | 1, 11 | syl 17 | 1 β’ (π β (βπ¦ β ((neiβπ½)βπ·)(π₯ β© π¦) = β β βπ¦ β π½ (π· β π¦ β§ (π₯ β© π¦) = β ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3074 β© cin 3914 β wss 3915 β c0 4287 βcfv 6501 Topctop 22258 neicnei 22464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-top 22259 df-nei 22465 |
This theorem is referenced by: sepnsepo 47030 |
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